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A Steiner bundle $E$ on ${\mathbb{P}}^n$ has a linear resolution of the form $0\to \mathcal{O}{\left(-1\right)}^s\to {\mathcal{O}}^t\to E\to 0$. In this paper we prove that a generic Steiner bundle $E$ is simple if and only if $\chi \left(\mathrm{End}E\right)$ is less or equal to 1. In particular we show that either $E$ is exceptional or it satisfies the inequality $t\le \left(\frac{n+1+\sqrt{{\left(n+1\right)}^2-4}}{2}\right)s$.

In the first part of the paper we complete the classification of the arithmetical Cohen-Macaulay vector bundles of rank 2 on a smooth prime Fano threefold. In the second part, we study some moduli spaces of these vector bundles, using the decomposition of the derived category of $X$ provided by Kuznetsov, when the genus of $X$ is 7 or 9. This allows to prove that such moduli spaces are birational to Brill-Noether varieties of vector bundles on a smooth projective curve $\mathrm{\Gamma }$. When the second Chern class...